A206586 Numbers k such that the periodic part of the continued fraction of sqrt(k) has positive even length.

3, 6, 7, 8, 11, 12, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33, 34, 35, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 80, 83, 84, 86, 87, 88, 90, 91, 92, 93, 94

Offset

1,1

Comments

By making the length positive, we exclude squares. See A206587 for this sequence and the squares. All primes of the form 4m + 3 are here.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

P. J. Rippon and H. Taylor, Even and odd periods in continued fractions of square roots, Fibonacci Quarterly 42, May 2004, pp. 170-180.

Mathematica

Select[Range[100], ! IntegerQ[Sqrt[#]] && EvenQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &]

Prog

(PARI)
cyc(cf) = {
  if(#cf==1, return(0)); \\ There is no cycle
  my(s=[]);
  for(k=2, #cf,
    s=concat(s, cf[k]);
    if(cf[k]==2*cf[1], return(s)) \\ Cycle found
  );
  0 \\ Cycle not found
}
select(n->(t=#cyc(contfrac(sqrt(n))))>0 && t%2==0, vector(100, n, n)) \\ Colin Barker, Oct 19 2014

Crossrefs

Cf. A003814 (period is odd), A206587.

Sequence in context: A047283 A155932 A298980 * A347878 A289176 A277851

Adjacent sequences: A206583 A206584 A206585 * A206587 A206588 A206589

Keyword

nonn

Author

T. D. Noe, Mar 19 2012

Status

approved